Skip to main content
Log in
Module 4: Electrons, waves, and photons›Series and parallel circuits (4.3.1)

Series and parallel circuits (4.3.1)

Combining resistors in series and parallel, Kirchhoff's laws, current and potential difference rules in circuit networks in A-level Physics.
5 min

On this page

Kirchhoff’s second law states that the sum of the electromotive force (EMF) is equal to the sum of the potential difference (p.d.) in a loop of a circuit.

Alternatively, the sum of the EMF and the p.d. in a loop of a circuit is equal to zero:

  • EMF is energy per unit charge (voltage) transferred from chemical to electrical energy. It is produced by an electrochemical cell or a changing magnetic field.
  • p.d. across a component is energy per unit charge (voltage) dissipated across the component.

So, this law is a consequence of the conservation of energy.

Add to favourites

In any circuit loop, potential rises across batteries or cells by their EMF and drops across components by their resistance multiplied by the current flowing through them.

A circuit diagram featuring a battery and two resistors. The battery, labeled 6V and ε = 6V, is connected in series with two resistors. The first resistor is labeled 1Ω, and the second is labeled 2Ω. Current arrows indicate a current of 2A flowing through the circuit. Annotations show calculations of potential differences across the resistors: 'p.d. = (2A)(1Ω) = 2V' for the 1Ω resistor and 'p.d. = (2A)(2Ω) = 4V' for the 2Ω resistor.
Add to favourites

The sum of the EMF is equal to the sum of the p.d. in any loop of a circuit.

If the direction of the current is negative in any part of a loop, then the direction of the p.d. is negative.

The image displays three circuit diagrams, each with a battery labeled 10 V and two resistors. In the top left circuit, arrows indicate current flow clockwise, with labels 'ε = 10 V' and 'p.d. = 10 V' shown. In the top right circuit, current flows counterclockwise, with labels 'p.d. = 10 V' and 'p.d. = -10 V'. In the bottom circuit, current flows clockwise, with labels 'ε = 10 V' and 'p.d. = -10 V'.

Only the red parts of the circuit are the loop being analysed in the examples above. Black sections are part of the real circuit but aren’t used in that loop’s equation.

Curved arrows show the direction you’re tracing the loop, not the actual current direction.

Trace with the current and the p.d. is positive; trace against it and the p.d. is negative, even though the resistor’s real p.d. hasn’t changed.

Add to favourites

Question walkthrough

Using Kirchhoff's Second Law to Find Voltage

Apply Kirchhoff's second law to a series circuit with a thermistor and filament lamp to find a missing voltmeter reading from the cell EMF and one known p.d.

Question walkthrough

Finding Potential Difference Between Two Points

Calculate the potential difference between two points in a circuit by finding the p.d. across each branch with Ohm's law and taking their difference.

Kirchhoff’s first law states that the sum of the current flowing into a junction is equal to the current flowing out of a junction.

A diagram illustrating Kirchhoff's first law with a horizontal line intersecting with three vertical lines forming a junction. The horizontal line is labeled 'I' and has an arrow pointing to the right. The three vertical lines connected at the junction are labeled 'I1', 'I2', and 'I3', with arrows pointing to the right for 'I1' and 'I2', and to the left for 'I3'. The text above reads: 'Kirchhoff’s first law: I = I1 + I2 + I3'.
Add to favourites

Question walkthrough

Applying Kirchhoff's Current Law at Junctions

Determine three unknown branch currents by applying Kirchhoff's current law at successive junctions in a multi-branch circuit.

Question walkthrough

Solving for Unknown Ammeter and Voltmeter Readings

Find ammeter and voltmeter readings in a series-parallel circuit using Ohm's law and current splitting across parallel resistors, with an alternative potential-divider method.

In a series circuit:

  • the current through each resistor is the same.
  • the sum of the voltages across the resistors is equal to the total voltage.
A circuit diagram showing a series circuit with a voltage source and three resistors. The voltage source is labeled 'V' with arrows indicating voltage across it. Three resistors are shown in sequence labeled as R1, R2, and R3. Arrows labeled V1, V2, and V3 indicate voltage across R1, R2, and R3 respectively. A current 'I' is shown entering and exiting the circuit. An equation V = V1 + V2 + V3 is annotated in the top right corner.
Add to favourites

The total resistance of two or more resistors in series is given by:

A circuit diagram showing three resistors in series labeled as R1, R2, and R3. There is a battery symbol with two parallel lines representing the power source. Below the resistors, the text reads: Total resistance = R1 + R2 + R3.
Add to favourites

Question walkthrough

Calculating Current in a Series Circuit

Find the current in a series circuit by summing three resistances to get the total resistance, then applying Ohm's law.

Question walkthrough

Deriving the Series Resistors Formula

Derive the total resistance formula for three resistors in series from Kirchhoff's voltage law and Ohm's law, showing each algebraic step.

In a parallel circuit:

  • the voltage across each resistor is the same,
  • the sum of the currents through the resistors is equal to the total current.
The image shows a parallel circuit diagram with a voltage source at the top labeled V. Three resistors, labeled R1, R2, and R3, are connected in parallel, each with a voltage drop labeled V across them. The current entering the circuit is labeled I. The current through each resistor is labeled I1 for R1, I2 for R2, and I3 for R3. An equation at the top right states I = I1 + I2 + I3. Arrows indicate the direction of current flow through the circuit.
Add to favourites

The total resistance of two or more resistors in parallel is given by:

The image shows a circuit diagram with three resistors labeled R1, R2, and R3 arranged in parallel. Each resistor is connected to the same two horizontal wires that are linked to a battery at the top of the diagram. On the right side is a mathematical expression for the total resistance in parallel: Total resistance = 1/R1 + 1/R2 + 1/R3. The image is © Medify.
Add to favourites

It is important to recall the correct formula for a circuit in parallel.

The image shows an electrical circuit diagram with a parallel configuration. The diagram includes two components labeled R2 and S1 connected in parallel. Above the components is a symbol representing a battery or power source. Below the circuit diagram, there is a mathematical expression: 1/R_total = 1/1Ω + 1/2Ω = 3/2Ω, followed by R_total = 2Ω/3.
Do

Use the formula:

to calculate the total resistance of resistors in parallel.

The image shows an electrical circuit diagram with a battery symbol at the top connected to two components in parallel. The first component is labeled 'R2' and the second component is labeled 'S1'. Below the diagram, there is an equation: R_total = 1Ω + 2Ω = 3Ω, indicating the total resistance calculation.
Don't

Confuse resistors in parallel with resistors in series by adding the resistances.

Add to favourites

Question walkthrough

Find total resistance in a circuit

By analysing and interpreting a circuit diagram, determine the total resistance in a circuit and the EMF of the power source.

Question walkthrough

Deriving the Parallel Resistors Formula

Derive the total resistance formula for three parallel resistors from Kirchhoff's current law and Ohm's law, showing each algebraic step.

Some circuits contain both series (denoted in the diagram as ) and parallel combinations. In such cases, the equations for series and parallel resistors can be combined.

The image shows two circuit diagrams with equations for total resistance. On the left, the equation '1/R_total = 1/(R_1 + R_2 + R_3) + 1/(S_1 + S_2)' is shown above a circuit. This circuit has three resistors in series labeled R_1, R_2, and R_3, followed by two resistors in parallel labeled S_1 and S_2. On the right, the equation 'R_total = R_1 + R_2 + 1/(1/S_1 + 1/S_2 + 1/S_3)' is above another circuit. This circuit has two resistors in series labeled R_1 and R_2, followed by three resistors in parallel labeled S_1, S_2, and S_3.
Add to favourites

Question walkthrough

Finding LED Resistance in a Mixed Circuit

Calculate an LED's resistance in a circuit combining series and parallel resistors, using Ohm's law and the reciprocal rule for parallel combinations.

Question walkthrough

Combining Three Parallel Resistor Branches

Practise finding total resistance for a network with three parallel branches, including a branch made of two series resistors, using the reciprocal formula for parallel combinations.

Question walkthrough

Finding Cell Current Using a Filament Graph

Uses a non-linear voltage-current graph for a filament lamp to find branch currents in a series-parallel circuit and the total current in the cell.

When analysing circuits with multiple cells, use Kirchhoff’s second law to form equations relating EMF, current and resistance.

  • Potential rises by across cells, where is the resistance of the cell.
  • Potential drops by across components (including internal resistors) where is the current flowing through the component and is its resistance.
Add to favourites

For two cells in series, the total EMF is equal to the sum of the EMF of each source.

If two cells of EMF, and point in the same direction, then the total EMF will be:

If two cells of EMF, and where point in opposite directions, then the total EMF will be:

Add to favourites

Connecting two cells of different EMF with no internal resistance in parallel will cause a dangerously high current to flow through the connecting wires, resulting in a short circuit. This occurs because there is a nonzero EMF (equal to the difference in EMF of the two cells) across a wire which has a very low resistance:

Therefore, large and small lead to a small For two cells of identical EMF in parallel, the total EMF remains

The image depicts a circuit diagram with two parallel branches. The top branch contains a single cell labeled with the symbol ε. The bottom branch also contains a single cell labeled with the symbol ε. Both cells are connected in parallel. The branches converge and connect to a rectangular component at the bottom of the diagram, likely representing a resistor. The diagram is labeled with © Medify at the bottom.
Add to favourites

Question walkthrough

Finding Current in Opposing Cells

Finds the magnitude and direction of current in a series circuit containing two cells connected in opposing directions.

Question walkthrough

Finding Thermistor Resistance from Ammeter Reading

Uses circuit symmetry with Kirchhoff's laws to find the current through a thermistor and calculate its resistance from an ammeter reading.

Question walkthrough

Solving Branch Currents with Kirchhoff's Laws

Applies Kirchhoff's first and second laws to a two-loop circuit to solve simultaneously for three unknown branch currents.

Question walkthrough

Resistance Effect on Meter Readings

Explains why increasing a variable resistor's resistance in a two-cell circuit has no effect on the voltmeter or ammeter readings across the fixed resistor.